Average Error: 59.7 → 2.3
Time: 2.6m
Precision: 64
Internal Precision: 1344
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[\frac{-\log \left((\frac{1}{12} \cdot \left(\pi \cdot f\right) + \left((\left(\left(\pi \cdot f\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(\left(\frac{1}{2880} \cdot f\right) \cdot \left(-f\right)\right) + \left(\frac{\frac{4}{f}}{\pi}\right))_*\right))_*\right)}{\frac{\pi}{4}}\]

Error

Bits error versus f

Derivation

  1. Initial program 59.7

    \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]

  2. Taylor expanded around 0 2.3

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\left(\frac{1}{12} \cdot \left(f \cdot \pi\right) + 4 \cdot \frac{1}{\pi \cdot f}\right) - \frac{1}{2880} \cdot \left({f}^{3} \cdot {\pi}^{3}\right)\right)}\]
  3. Using strategy rm

  4. Applied associate-*l/2.2

    \[\leadsto -\color{blue}{\frac{1 \cdot \log \left(\left(\frac{1}{12} \cdot \left(f \cdot \pi\right) + 4 \cdot \frac{1}{\pi \cdot f}\right) - \frac{1}{2880} \cdot \left({f}^{3} \cdot {\pi}^{3}\right)\right)}{\frac{\pi}{4}}}\]

  5. Simplified2.3

    \[\leadsto -\frac{\color{blue}{\log \left((\frac{1}{12} \cdot \left(f \cdot \pi\right) + \left((\left(\left(\pi \cdot \pi\right) \cdot \left(f \cdot \pi\right)\right) \cdot \left(\left(\frac{1}{2880} \cdot f\right) \cdot \left(-f\right)\right) + \left(\frac{\frac{4}{f}}{\pi}\right))_*\right))_*\right)}}{\frac{\pi}{4}}\]

  6. Final simplification2.3

    \[\leadsto \frac{-\log \left((\frac{1}{12} \cdot \left(\pi \cdot f\right) + \left((\left(\left(\pi \cdot f\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(\left(\frac{1}{2880} \cdot f\right) \cdot \left(-f\right)\right) + \left(\frac{\frac{4}{f}}{\pi}\right))_*\right))_*\right)}{\frac{\pi}{4}}\]

Runtime

Time bar (total: 2.6m)Debug logProfile

herbie shell --seed 2018242 +o rules:numerics
(FPCore (f)
  :name "VandenBroeck and Keller, Equation (20)"
  (- (* (/ 1 (/ PI 4)) (log (/ (+ (exp (* (/ PI 4) f)) (exp (- (* (/ PI 4) f)))) (- (exp (* (/ PI 4) f)) (exp (- (* (/ PI 4) f)))))))))